Problem: If $\sqrt{5 + x} + \sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$?
Square both sides of the equation to obtain \[
(5+x)+2\sqrt{5+x}\sqrt{20-x}+(20-x)=49.
\]This equation simplifies to \[
2\sqrt{(5+x)(20-x)}=24,
\]so $(5+x)(20-x)=(24/2)^2=\boxed{144}$.